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Chapter 2 2 1 1

1.
2.4 From Regular Expression To DFAs
An algorithm: translating a regular expression into a DFA via NFA.
regular expression NFA DFA program
2.4.1 From a regular expression to an NFA
The idea of thompson’s construction:
Use ε-transitions to “glue together” the machine of each piece of a regular
expression to form a machine that corresponds to the whole expression.
1. Basic regular expression:
a basic regular expression is of the form a, ε,or φ
a
ε
2. Concatenation: rs
ε
r s
¨¨ ¨¨
We have connected the accepting state of the machine of r to the start state of the
machine of s by anε-transition.
The new machine has the start stale of the machine of r as its start state and the
accepting state of the machine of j as its accepting state.
Clearly, this machine accepts L(rs) = L(r)L(s) and so corresponds to the regular
expression rs.
3. Choice among alternatives: r|s
ε r
¨¨ ε
s
ε ε
¨¨
We have added a new start state and a new accepting state and connected them as
shown usingε-transitions.
Clearly, this machine accepts the language L(r|s) =L(r )UL ( s).
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2.
4. Repetition: construct a machine that corresponds to r*
Given a machine that corresponds to r.
We added two new states, a start state and an accepting state.
The repetition in this machine is afforded by the newε-transition from the
accepting state of the machine of r to its start state.
We most also draw an ε-transition from the new start state to the new
accepting state.
This construction is not unique, simplifications are possible in the many cases.
Example 1: The regular expression ab|a
a
b
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3.
Example 2: regular expression letter(letter|digit)*
The machine for the choice letter |digit:
letter digit
The NFA for the repetition (letter|digit) * as follows:
ε
Finally, the NFA for the letter(letter|digit)*: ε
2.4.2 from an NFA to a DFA
Given an arbitrary NFA, construct an equivalent DFA. (i.e., one that accepts
precisely the same strings)
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4.
We need some methods for:
(1) Eliminating ε-transitions
ε-closure: the set of all states reachable by ε-transitions from a state or states.
(2) Eliminating multiple transitions from a state on a single input character.
Keeping track of the set of states that are reachable by matching a single
character.
Both these processes lead us to consider sets of states instead of single states.
Thus, it is not surprising that the DFA we construct has as its states sets of states of the
original NFA.
The algorithm is called the subset construction.
1. The ε-closure of a Set of states:
The ε-closure of a single state s is the set of states reachable by a series of zero or
more ε-transitions, and we write this set as s .
Example 2.14: regular a*
1 = { 1，2，4}， 2 ={2}， 3 ={2，3，4}， and 4 ={4}.
The ε-closure of a set of states : the union of the ε-closures of each individual state.
S= s
sin S
{1,3} = 1 ∪ 3 = {1，2，3}∪{2，3，4}={1，2，3，4}
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5.
2. The Subset Construction:
(1) compute the ε-closure of the start state of M; this becomes the start state of M
.
(2) For this set, and for each subsequent set, we compute transitions on characters
a as follows.
Given a set S of states and a character a in the alphabet,
Compute the set
S′a = { t | for some s in S there is a transition from s to t on a }.
Then, compute S a ' , the ε-closure of S′a.
This defines a new state in the subset construction, together with a new
transition S→ S a ' .
(3) Continue with this process until no new states or transitions are created.
(4) Mark as accepting those states constructed in this manner that contain an
accepting state of M.
Example 2.15: consider the NFA of example 2.14
M ε-closure of M ( S ) S′a
1 1,2,4 2,3,4
2,3,4 2,3,4 2,3,4
a a
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